I like Mathematica and find it a sometimes, very useful tool. I've taken over two dozen University courses in mathematics, including topology, abstract algebra, analysis, combinatorics, number theory, etc. Yet, even simply equations can stump my rather meager ability to solve them and this is where Mathematica is so useful. You can use it interactively to quickly traverse the difficult ground on the way to insight or a solution about your problems.
Consider the simple looking problem, find the definite integral of sin(x^2) from 0 to positive infinity. This is a real expression that that occurs naturally in the study of electromagnetism.
Solving this is beyond the ability of most practicing mathematicians to just sit down and noodle out an answer. The integral in known as the Fresnel integral and has the pretty answer of sqrt(pi/2)/2. Mathematica gives me the answer in just a second.
Yes, I've tried SageMath, which is said to build on sympy as well as numpy, scipy, R, maxima, and other math related projects. It's really quite capable considering that it is an open source project. It supports notebooks, like Mathematica, and the programming is done in Python, which is much more widely known that the Mathematica programming language. I like it's wide open nature and the programming in Python. However, there are still some rough spots where the different technologies come together. Mathematica on the other hand is expensive but it is very polished.
SageMath, http://www.sagemath.org, is perfect for those that can't afford Mathematica or want to leverage their existing knowledge of Python. It solves the Fresnel Integral too. Please consider supporting the SageMath project.
Consider the simple looking problem, find the definite integral of sin(x^2) from 0 to positive infinity. This is a real expression that that occurs naturally in the study of electromagnetism.
Solving this is beyond the ability of most practicing mathematicians to just sit down and noodle out an answer. The integral in known as the Fresnel integral and has the pretty answer of sqrt(pi/2)/2. Mathematica gives me the answer in just a second.